I am a graduate student and want to study the theory of currents. What is a good reference for a beginner? I should be familiar with the theory of distributions or generalized functions on $\mathbb R^n$.
The theory of currents is a part of the geometric measure theory. Unfortunately, Federer made the subject completely inaccessible after he wrote his famous monograph:
H. Federer, Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969.
The problem is that the book contains `everything' (well, almost) and it is unreadable. After this book was published, people did not dare to write other books on the topic and only the bravest hearts dared to read Federer's Bible.
In my opinion the first accessible book on the subject is
L. Simon, Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, 3. Australian National University, Centre for Mathematical Analysis, Canberra, 1983.
You can find it as a pdf file in the internet. Note that this book was written 14 years after Federer's book and there was nothing in between.
I would also suggest:
F. Lin, X. Yang, Geometric measure theory—an introduction. Advanced Mathematics (Beijing/Boston), 1. Science Press Beijing, Beijing; International Press, Boston, MA, 2002.
I haven't read it, but it looks relatively elementary (relatively, because by no means the subject is elementary).
The last, but not least is
F. Morgan, Geometric measure theory. A beginner's guide. Fifth edition. Illustrated by James F. Bredt. Elsevier/Academic Press, Amsterdam, 2016.
You will not learn anything form that book as it does not have detailed proofs, but you can read it rather quickly and after that you will have an idea about what it is all about.